the phase structure of CDT

In my previous post I criticized the description of the CDT phase diagram on a popular physics blog.
In this post I want to actually talk about the numerical CDT results.

The phase diagram depends on two coupling constants K and D (in the text they use kappa and delta). While K corresponds to the gravitational coupling, D measures the ratio of 'timelike' and 'spacelike' edges; I use quotes ' ' because the simulation is actually done in the Euclidean sector, but edges fall in different categories, depending on what kind of distances they would correspond to after Wick rotation. There is a third coupling parameter, which corresponds to a cosmological constant, but it is fixed for technical reasons.

As I already explained, one looks for a critical line in D,K corresponding to a 2nd order phase transition and the reason is that long-range fluctuations are associated with such a transition, so that the details of the discretization do not matter any more.
So this is what I find weird: The parameter D describes a detail of the discrete model and the hope is to fine tune D, as a function of K, in order to find a critical line where the details of the discretization no longer matter...

The authors notice that D has "no immediate interpretation in the Einstein-Hilbert action" and thus the critical value D(K) does not correspond to any feature of the continuum limit - unless the continuum limit is not Einstein-Hilbert but Horava-Lifshitz gravity. This is what the authors propose and discuss in section 4 of their paper: HL gravity breaks diffeomorphism invariance of EH gravity, just like CDT does, and the parameter D would have a 'physical' meaning in this case.

It seems that the authors hope that EH gravity will be restored somewhere along the critical D(K) line, however, it is unlikely imho that there is such a path from HL gravity to real gravitation.